PREGEL. A System for Large Scale Graph Processing

Transcription

1 PREGEL A System for Large Scale Graph Processing

2 The Problem Large Graphs are often part of computations required in modern systems (Social networks and Web graphs etc.) There are many graph computing problems like shortest path, clustering, page rank, minimum cut, connected components etc. but there exists no scalable general purpose system for implementing them. Pregel 2

3 Characteristics of the algorithms They often exhibit poor locality of memory access. Very little computation work required per vertex. Changing degree of parallelism over the course of execution. Pregel Refer [1, 2] 3

4 Possible solutions Crafting a custom distributed framework for every new algorithm. Existing distributed computing platforms like MapReduce. These are sometimes used to mine large graphs [3, 4], but often give sub optimal performance and have usability issues. Single computer graph algorithm libraries Limiting the scale of the graph is necessary BGL, LEDA, NetworkX, JDSL, Standford GraphBase or FGL Existing parallel graph systems which do not handle fault tolerance and other issues The Parallel BGL [5] and CGMgraph [6] Pregel 4

5 Pregel Google, to overcome, these challenges came up with Pregel. Provides scalability Fault tolerance Flexibility to express arbitrary algorithms The high level organization of Pregel programs is inspired by Valiant s Bulk Synchronous Parallel model [7]. Pregel 5

6 Message passing model A pure message passing model has been used, omitting remote reads and ways to emulate shared memory because: 1. Message passing model was found sufficient for all graph algorithms 2. Message passing model performs better than reading remote values because latency can be amortized by delivering larges batches of messages asynchronously. Pregel 6

7 Message passing model Pregel 7

8 Example Find the largest value of a vertex in a strongly connected graph Pregel 8

9 Finding the largest value in a graph Blue Arrows are messages Blue vertices have voted to halt Pregel 9

10 Basic Organization Computations consist of a sequence of iterations called supersteps. During a superstep, the framework invokes a user defined function for each vertex which specifies the behavior at a single vertex V and a single Superstep S. The function can: Read messages sent to V in superstep S 1 Send messages to other vertices that will be received in superstep S+1 Modify the state of V and of the outgoing edges Make topology changes (Introduce/Delete/Modify edges/vertices) Pregel 10

11 Basic Organization - Superstep Pregel 11

12 Model Of Computation: Entities VERTEX Identified by a unique identifier. Has a modifiable, user defined value. EDGE Source vertex and Target vertex identifiers. Has a modifiable, user defined value. Pregel 12

13 Model Of Computation: Progress In superstep 0, all vertices are active. Only active vertices participate in a superstep. They can go inactive by voting for halt. They can be reactivated by an external message from another vertex. The algorithm terminates when all vertices have voted for halt and there are no messages in transit. Pregel 13

14 Model Of Computation: Vertex State machine for a vertex Pregel 14

15 Comparison with MapReduce Graph algorithms can be implemented as a series of MapReduce invocations but it requires passing of entire state of graph from one stage to the next, which is not the case with Pregel. Also Pregel framework simplifies the programming complexity by using supersteps. Pregel 15

16 The C++ API Creating a Pregel program typically involves subclassing the predefined Vertex class. The user overrides the virtual Compute() method. This method is the function that is computed for every active vertex in supersteps. Compute() can get the vertex s associated value by GetValue() or modify it using MutableValue() Values of edges can be inspected and modified using the out edge iterator. Pregel 16

17 The C++ API Message Passing Each message consists of a value and the name of the destination vertex. The type of value is specified in the template parameter of the Vertex class. Any number of messages can be sent in a superstep. The framework guarantees delivery and nonduplication but not in order delivery. A message can be sent to any vertex if it s identifier is known. Pregel 17

18 The C++ API Pregel Code Pregel Code for finding the max value Class MaxFindVertex : public Vertex<double, void, double> { public: virtual void Compute(MessageIterator* msgs) { int currmax = GetValue(); SendMessageToAllNeighbors(currMax); for ( ;!msgs >Done(); msgs >Next()) { if (msgs >Value() > currmax) currmax = msgs >Value(); } if (currmax > GetValue()) *MutableValue() = currmax; else VoteToHalt(); } }; Pregel 18

19 The C++ API Combiners Sending a message to another vertex that exists on a different machine has some overhead. However if the algorithm doesn t require each message explicitly but a function of it (example sum) then combiners can be used. This can be done by overriding the Combine() method. It can be used only for associative and commutative operations. Pregel 19

20 The C++ API Combiners Example: Say we want to count the number of incoming links to all the pages in a set of interconnected pages. In the first iteration, for each link from a vertex(page) we will send a message to the destination page. Here, count function over the incoming messages can be used a combiner to optimize performance. In the MaxValue Example, a Max combiner would reduce the communication load. Pregel 20

21 The C++ API Combiners Pregel 21

22 The C++ API Aggregators They are used for Global communication, monitoring and data. Each vertex can produce a value in a superstep S for the Aggregator to use. The Aggregated value is available to all the vertices in superstep S+1. Aggregators can be used for statistics and for global communication. Can be implemented by subclassing the Aggregator Class Commutativity and Assosiativity required Pregel 22

23 The C++ API Aggregators Example: Sum operator applied to out edge count of each vertex can be used to generate the total number of edges in the graph and communicate it to all the vertices. More complex reduction operators can even generate histograms. In the MaxValue example, we can finish the entire program in a single superstep by using a Max aggregator. Pregel 23

24 The C++ API Topology Mutations The Compute() function can also be used to modify the structure of the graph. Example: Hierarchical Clustering Mutations take effect in the superstep after the requests were issued. Ordering of mutations, with deletions taking place before additions, deletion of edges before vertices and addition of vertices before edges resolves most of the conflicts. Rest are handled by user defined handlers. Pregel 24

25 Implementation Pregel is designed for the Google cluster architecture. The architecture schedules jobs to optimize resource allocation, involving killing instances or moving them to different locations. Persistent data is stored as files on a distributed storage system like GFS [8] or BigTable. Pregel 25

26 Basic Architecture The Pregel library divides a graph into partitions, based on the vertex ID, each consisting of a set of vertices and all of those vertices out going edges. The default function is hash(id) mod N, where N is the number of partitions. The next few slides describe the several stages of the execution of a Pregel program. Pregel 26

27 Pregel Execution 1. Many copies of the user program begin executing on a cluster of machines. One of these copies acts as the master. The master is not assigned any portion of the graph, but is responsible for coordinating worker activity. Pregel 27

28 Pregel Execution 2. The master determines how many partitions the graph will have and assigns one or more partitions to each worker machine. Each worker is responsible for maintaining the state of its section of the graph, executing the user s Compute() method on its vertices, and managing messages to and from other workers. Pregel 28

29 Pregel Execution Pregel 29

30 Pregel Execution 3. The master assigns a portion of the user s input to each worker. The input is treated as a set of records, each of which contains an arbitrary number of vertices and edges. After the input has finished loading, all vertices are marked are active. Pregel 30

31 Pregel Execution 4. The master instructs each worker to perform a superstep. The worker loops through its active vertices, and call Compute() for each active vertex. It also delivers messages that were sent in the previous superstep. When the worker finishes it responds to the master with the number of vertices that will be active in the next superstep. Pregel 31

32 Pregel Execution Pregel 32

33 Pregel Execution Pregel 33

34 Fault Tolerance Checkpointing is used to implement fault tolerance. At the start of every superstep the master may instruct the workers to save the state of their partitions in stable storage. This includes vertex values, edge values and incoming messages. Master uses ping messages to detect worker failures. Pregel 34

35 Fault Tolerance When one or more workers fail, their associated partitions current state is lost. Master reassigns these partitions to available set of workers. They reload their partition state from the most recent available checkpoint. This can be many steps old. The entire system is restarted from this superstep. Confined recovery can be used to reduce this load Pregel 35

36 Applications PageRank Pregel 36

37 PageRank PageRank is a link analysis algorithm that is used to determine the importance of a document based on the number of references to it and the importance of the source documents themselves. [This was named after Larry Page (and not after rank of a webpage)] Pregel 37

38 PageRank A = A given page T 1. T n = Pages that point to page A (citations) d = Damping factor between 0 and 1 (usually kept as 0.85) C(T) = number of links going out of T PR(A) = the PageRank of page A PR( A) (1 d) d ( PR( T1 ) C( T ) 1 PR( T2 ) C( T ) 2... PR( Tn ) ) C( T ) n Pregel 38

39 PageRank Courtesy: Wikipedia Pregel 39

40 PageRank PageRank can be solved in 2 ways: A system of linear equations An iterative loop till convergence We look at the pseudo code of iterative version Initial value of PageRank of all pages = 1.0; While ( sum of PageRank of all pages numpages > epsilon) { for each Page P i in list { PageRank(P i ) = (1 d); for each page P j linking to page P i { PageRank(P i ) += d (PageRank(P j )/numoutlinks(p j )); } } } Pregel 40

41 PageRank in MapReduce Phase I Parsing HTML Map task takes (URL, page content) pairs and maps them to (URL, (PR init, list of urls)) PR init is the seed PageRank for URL list of urls contains all pages pointed to by URL Reduce task is just the identity function Pregel 41

42 PageRank in MapReduce Phase 2 PageRank Distribution Map task takes (URL, (cur_rank, url_list)) For each u in url_list, emit (u, cur_rank/ url_list ) Emit (URL, url_list) to carry the points to list along through iterations Reduce task gets (URL, url_list) and many (URL, val) values Sum vals and fix up with d Emit (URL, (new_rank, url_list)) Pregel 42

43 PageRank in MapReduce - Finalize A non parallelizable component determines whether convergence has been achieved If so, write out the PageRank lists done Otherwise, feed output of Phase 2 into another Phase 2 iteration Pregel 43

44 PageRank in Pregel Class PageRankVertex : public Vertex<double, void, double> { public: virtual void Compute(MessageIterator* msgs) { if (superstep() >= 1) { double sum = 0; for (;!msgs >done(); msgs >Next()) sum += msgs >Value(); *MutableValue() = * sum; } if (supersteps() < 30) { const int64 n = GetOutEdgeIterator().size(); SendMessageToAllNeighbors(GetValue() / n); } else { VoteToHalt(); }}}; Pregel 44

45 PageRank in Pregel The pregel implementation contains the PageRankVertex, which inherits from the Vertex class. The class has the vertex value type double to store tentative PageRank and message type double to carry PageRank fractions. The graph is initialized so that in superstep 0, value of each vertex is 1.0. Pregel 45

46 PageRank in Pregel In each superstep, each vertex sends out along each outgoing edge its tentative PageRank divided by the number of outgoing edges. Also, each vertex sums up the values arriving on messages into sum and sets its own tentative PageRank to sum For convergence, either there is a limit on the number of supersteps or aggregators are used to detect convergence. Pregel 46

47 Apache Giraph Large-scale Graph Processing on Hadoop Claudio Hadoop Amsterdam - 3 April 2014

48 2

49 Graphs are simple 3

50 A computer network 4

51 A social network 5

52 A semantic network 6

53 A map 7

54 Graphs are huge Google s index contains 50B pages Facebook has around1.1b users Google+ has around 570M users Twitter has around 530M users VERY rough estimates! 8

55 9

56 Graphs aren t easy 10

57 Graphs are nasty. 11

58 Each vertex depends on its neighbours, recursively. 12

59 Recursive problems are nicely solved iteratively. 13

60 PageRank in MapReduce Record: < v_i, pr, [ v_j,..., v_k ] > Mapper: emits < v_j, pr / #neighbours > Reducer: sums the partial values 14

61 MapReduce dataflow 15

62 Drawbacks Each job is executed N times Job bootstrap Mappers send PR values and structure Extensive IO at input, shuffle & sort, output 16

63 17

64 Timeline Inspired by Google Pregel (2010) Donated to ASF by Yahoo! in 2011 Top-level project in release in January release in days

65 Plays well with Hadoop 19

66 Vertex-centric API 20

67 BSP machine 21

68 BSP & Giraph 22

69 Advantages No locks: message-based communication No semaphores: global synchronization Iteration isolation: massively parallelizable 23

70 Architecture 24

71 Giraph job lifetime 25

72 Designed for iterations Stateful (in-memory) Only intermediate values (messages) sent Hits the disk at input, output, checkpoint Can go out-of-core 26

73 A bunch of other things Combiners (minimises messages) Aggregators (global aggregations) MasterCompute (executed on master) WorkerContext (executed per worker) PartitionContext (executed per partition) 27

74 Shortest Paths 28

75 Shortest Paths 29

76 Shortest Paths 30

77 Shortest Paths 31

78 Shortest Paths 32

79 Composable API 33

80 Checkpointing 34

81 No SPoFs 35

82 Giraph scales ref: dges/

83 Giraph is fast 100x over MR (Pr) jobs run within minutes given you have resources ;-) 37

84 Serialised objects 38

85 Primitive types Autoboxing is expensive Objects overhead (JVM) Use primitive types on your own Use primitive types-based libs (e.g. fastutils) 39

86 Sharded aggregators 40

87 Many stores with Gora 41

88 And graph databases 42

89 Current and next steps Out-of-core graph and messages Jython interface Remove Writable from < I V E M > Partitioned supernodes More documentation 43

90 GraphLab: A New Framework for Parallel Machine Learning Yucheng Low, Aapo Kyrola, Carlos Guestrin, Joseph Gonzalez, Danny Bickson, Joe Hellerstein Presented by Guozhang Wang DB Lunch, Nov.8, 2010

91 Overview Programming ML Algorithms in Parallel Common Parallelism and MapReduce Global Synchronization Barriers GraphLab Data Dependency as a Graph Synchronization as Fold/Reduce Implementation and Experiments From Multicore to Distributed Environment

92 Parallel Processing for ML Parallel ML is a Necessity 13 Million Wikipedia Pages 3.6 Billion photos on Flickr etc Parallel ML is Hard to Program Concurrency v.s. Deadlock Load Balancing Debug etc

93 MapReduce is the Solution? High-level abstraction: Statistical Query Model [Chu et al, 2006] Weighted Linear Regression: only sufficient statistics Θ = A -1 b, A = Σw i (x i x it ), b = Σw i (x i y i )

94 MapReduce is the Solution? High-level abstraction: Statistical Query Model [Chu et al, 2006] Embarrassingly K-Means: Parallel only independent data assignments computation No Communication class mean needed = avg(x i ), x i in class

95 ML in MapReduce Single Reducer Multiple Mapper Iterative MapReduce needs global synchronization at the single reducer K-means EM for graphical models gradient descent algorithms, etc

96 Not always Embarrassingly Parallel Data Dependency: not MapReducable Gibbs Sampling Belief Propagation SVM etc Capture Dependency as a Graph!

97 Overview Programming ML Algorithms in Parallel Common Parallelism and MapReduce Global Synchronization Barriers GraphLab Data Dependency as a Graph Synchronization as Fold/Reduce Implementation and Experiments From Multicore to Distributed Environment

98 Key Idea of GraphLab Sparse Data Dependencies Local Computations X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9

99 GraphLab for ML High-level Abstract Express data dependencies Iterative Automatic Multicore Parallelism Data Synchronization Consistency Scheduling

100 Main Components of GraphLab Data Graph Shared Data Table GraphLab Model Scheduling Update Functions and Scopes

101 Data Graph A Graph with data associated with every vertex and edge. X 1 X 2 X 3 X 4 x 3 : Sample value C(X 3 ): sample counts X 5 X 6 X 7 X X X 10 X 11 Φ(X 6,X 9 ): Binary potential 8 9

102 Update Functions Operations applied on a vertex that transform data in the scope of the vertex Gibbs Update: - Read samples on adjacent vertices - Read edge potentials - Compute a new sample for the current vertex

103 Scope Rules Consistency v.s. Parallelism Belief Propagation: Only uses edge data Gibbs Sampling: Needs to read adjacent vertices

104 Scheduling Scheduler determines the order of Update Function evaluations Static Scheduling Round Robin, etc Dynamic Scheduling FIFO, Priority Queue, etc

105 Dynamic Scheduling a CPU 1 a b c d h a e f g b h i j k CPU 2

106 Global Information Shared Data Table in Shared Memory Model parameters (updatable) Sufficient statistics (updatable) Constants, etc (fixed) Sync Functions for Updatable Shared Data Accumulate performs an aggregation over vertices Apply makes a final modification to the accumulated data

107 Sync Functions Much like Fold/Reduce Execute Aggregate over every vertices in turn Execute Apply once at the end Can be called Periodically when update functions are active (asynchronous) or By the update function or user code (synchronous)

108 GraphLab Data Graph Shared Data Table GraphLab Model Scheduling Update Functions and Scopes

109 Overview Programming ML Algorithms in Parallel Common Parallelism and MapReduce Global Synchronization Barriers GraphLab Data Dependency as a Graph Synchronization as Fold/Reduce Implementation and Experiments From Multicore to Distributed Environment

110 Implementation and Experiments Shared Memory Implemention in C++ using Pthreads Applications: Belief Propagation Gibbs Sampling CoEM Lasso etc (more on the project page)

111 Speedup Better Parallel Performance Optimal Colored Schedule Round robin schedule Number of CPUs

112 From Multicore to Distributed Enviroment MapReduce and GraphLab work well for Multicores Simple High-level Abstract Local computation + global synchronization When Migrate to Clusters Rethink Scope synchronization Rethink Shared Data single reducer Think Load Balancing Maybe think abstract model?

113 Hot Topics in Information Management PowerGraph: Distributed Graph-Parallel Computation on Natural Graphs Igor Shevchenko Mentor: Sebastian Schelter Fachgebiet Datenbanksysteme und Informationsmanagement Technische Universität Berlin DIMA TU Berlin 1

114 Agenda 1. Natural Graphs: Properties and Problems; 2. PowerGraph: Vertex Cut and Vertex Programs; 3. GAS Decomposition; 4. Vertex Cut Partitioning; 5. Delta Caching; 6. Applications and Evaluation; Paper: Gonzalez at al. PowerGraph: Distributed Graph- Parallel Computation on Natural Graphs DIMA TU Berlin 2

115 Natural Graphs Natural graphs are graphs derived from real-world or natural phenomena; Graphs are big: billions of vertices and edges and rich metadata; Natural graphs have Power-Law Degree Distribution DIMA TU Berlin 3

116 Power-Law Degree Distribution (Andrei Broder et al. Graph structure in the web) DIMA TU Berlin 4

117 Goal We want to analyze natural graphs; Essential for Data Mining and Machine Learning; Identify influential people and information; Identify special nodes and communities; Model complex data dependencies; Target ads and products; Find communities; Flow scheduling; DIMA TU Berlin 5

118 Problem Existing distributed graph computation systems perform poorly on natural graphs (Gonzalez et al. OSDI 12); The reason is presence of high degree vertices; High Degree Vertices: Star like motif DIMA TU Berlin 6

119 Problem Continued Possible problems with high degree vertices: Limited single-machine resources; Work imbalance; Sequential computation; Communication costs; Graph partitioning; Applicable to: Hadoop; GraphLab; Pregel (Piccolo); DIMA TU Berlin 7

120 Problem: Limited Single-Machine Resources High degree vertices can exceed the memory capacity of a single machine; Store edge meta-data and adjacency information; DIMA TU Berlin 8

121 Problem: Work Imbalance The power-law degree distribution can lead to significant work imbalance and frequency barriers; For ex. with synchronous execution (Pregel): DIMA TU Berlin 9

122 Problem: Sequential Computation No parallelization of individual vertex-programs; Edges are processed sequentially; Locking does not scale well to high degree vertices (for ex. in GraphLab); Sequentially process edges Asynchronous execution requires heavy locking DIMA TU Berlin 10

123 Problem: Communication Costs Generate and send large amount of identical messages (for ex. in Pregel); This results in communication asymmetry; DIMA TU Berlin 11

124 Problem: Graph Partitioning Natural graphs are difficult to partition; Pregel and GraphLab use random (hashed) partitioning on natural graphs thus maximizing the network communication; DIMA TU Berlin 12

125 Problem: Graph Partitioning Continued Natural graphs are difficult to partition; Pregel and GraphLab use random (hashed) partitioning on natural graphs thus maximizing the network communication; Expected edges that are cut = number of machines Examples: 10 machines: 100 machines: 90% of edges cut; 99% of edges cut; DIMA TU Berlin 13

126 In Summary GraphLab and Pregel are not well suited for computations on natural graphs; Reasons: Challenges of high-degree vertices; Low quality partitioning; Solution: PowerGraph new abstraction; DIMA TU Berlin 14

127 PowerGraph DIMA TU Berlin 15

128 Partition Techniques Two approaches for partitioning the graph in a distributed environment: Edge Cut; Vertex Cut; DIMA TU Berlin 16

129 Edge Cut Used by Pregel and GraphLab abstractions; Evenly assign vertices to machines; DIMA TU Berlin 17

130 Vertex Cut The strong point of the paper Used by PowerGraph abstraction; Evenly assign edged to machines; 4 edges 4 edges DIMA TU Berlin 18

131 Vertex Programs Think like a Vertex [Malewicz et al. SIGMOD 10] User-defined Vertex-Program: 1. Runs on each vertex; 2. Interactions are constrained by graph structure; Pregel and GraphLab also use this concept, where parallelism is achieved by running multiple vertex programs simultaneously; DIMA TU Berlin 19

132 GAS Decomposition The strong point of the paper Vertex cut distributes a single vertex-program across several machines; Allows to parallelize high-degree vertices; DIMA TU Berlin 20

133 GAS Decomposition The strong point of the paper Generalize the vertex-program into three phases: 1. Gather Accumulate information about neighborhood; 2. Apply Apply accumulated value to center vertex; 3. Scatter Update adjacent edges and vertices; Gather, Apply and Scatter are user defined functions; DIMA TU Berlin 21

134 Gather Phase Executed on the edges in parallel; Accumulate information about neighborhood; DIMA TU Berlin 22

135 Apply Phase Executed on the central vertex; Apply accumulated value to center vertex; DIMA TU Berlin 23

136 2D Partitioning Aydin Buluc and Kamesh Madduri 8 / 13

137 Graph Partitioning for Scalable Distributed Graph Computations Aydın Buluç Kamesh Madduri 10 th DIMACS Implementation Challenge, Graph Partitioning and Graph Clustering February 13 14, 2012 Atlanta, GA

138 Overview of our study We assess the impact of graph partitioning for computations on low diameter graphs Does minimizing edge cut lead to lower execution time? We choose parallel Breadth First Search as a representative distributed graph computation Performance analysis on DIMACS Challenge instances 2

139 Key Observations for Parallel BFS Well balanced vertex and edge partitions do not guarantee load balanced execution, particularly for real world graphs Range of relative speedups (8.8 50X, 256 way parallel concurrency) for low diameter DIMACS graph instances. Graph partitioning methods reduce overall edge cut and communication volume, but lead to increased computational load imbalance Inter node communication time is not the dominant cost in our tuned bulk synchronous parallel BFS implementation 3

140 Talk Outline Level synchronous parallel BFS on distributedmemory systems Analysis of communication costs Machine independent counts for inter node communication cost Parallel BFS performance results for several large scale DIMACS graph instances 4

141 Parallel BFS strategies 1. Expand current frontier (level synchronous approach, suited for low diameter graphs) source vertex O(D) parallel steps Adjacencies of all vertices in current frontier are visited in parallel 2. Stitch multiple concurrent traversals (Ullman Yannakakis, for high diameter graphs) source vertex path limited searches from super vertices APSP between super vertices 2 5

142 2D graph distribution Consider a logical 2D processor grid (p r * p c = p) and the dense matrix representation of the graph Assign each processor a sub matrix (i.e, the edges within the sub matrix) Flatten Sparse matrices Per processor local graph representation 9 vertices, 9 processors, 3x3 processor grid x x x x x x x x x x x x x x x x x x x x x x x x

143 BFS with a 1D partitioned graph Consider an undirected graph with n vertices and m edges 0 1 Each processor owns n/p vertices and stores their adjacencies (~ 2m/p per processor, assuming balanced partitions). [0,1] [0,3] [0,3] [1,0] [1,4] [1,6] [2,3] [2,5] [2,5] [2,6] [3,0] [3,0] [3,2] [3,6] [4,1] [4,5] [4,6] [5,2] [5,2] [5,4] [6,1] [6,2] [6,3] [6,4] 2 5 Steps: 1. Local discovery: Explore adjacencies of vertices in current frontier. 2. Fold: All to all exchange of adjacencies. 3. Local update: Update distances/parents for unvisited vertices.

144 BFS with a 1D partitioned graph 0 1 Current frontier: vertices 1 (partition Blue) and 6 (partition Green) P0 1. Local discovery: [1,0] [1,4] [1,6] P1 No work 2 5 P2 No work P3 [6,1] [6,2] [6, 3] [6,4] Steps: 1. Local discovery: Explore adjacencies of vertices in current frontier. 2. Fold: All to all exchange of adjacencies. 3. Local update: Update distances/parents for unvisited vertices.

145 BFS with a 1D partitioned graph 0 1 Current frontier: vertices 1 (partition Blue) and 6 (partition Green) P0 2. All to all exchange: [1,0] [1,4] [1,6] P1 No work 2 5 P2 No work P3 [6,1] [6,2] [6, 3] [6,4] Steps: 1. Local discovery: Explore adjacencies of vertices in current frontier. 2. Fold: All to all exchange of adjacencies. 3. Local update: Update distances/parents for unvisited vertices.

146 BFS with a 1D partitioned graph 0 1 Current frontier: vertices 1 (partition Blue) and 6 (partition Green) P0 2. All to all exchange: [1,0] [6,1] 2 5 P1 P2 P3 [6,2] [6, 3] [6,4] [1,6] [1,4] Steps: 1. Local discovery: Explore adjacencies of vertices in current frontier. 2. Fold: All to all exchange of adjacencies. 3. Local update: Update distances/parents for unvisited vertices.

147 BFS with a 1D partitioned graph P0 Current frontier: vertices 1 (partition Blue) and 6 (partition Green) 3. Local update: [1,0] [6,1] Frontier for next iteration P1 P2 P3 [6,2] [6, 3] [6,4] [1,6] [1,4] Steps: 1. Local discovery: Explore adjacencies of vertices in current frontier. 2. Fold: All to all exchange of adjacencies. 3. Local update: Update distances/parents for unvisited vertices. 2, 3 4

148 Modeling parallel execution time Time dominated by local memory references and inter node communication Assuming perfectly balanced computation and communication, we have Local memory references: L m p L, n / p n m p Inter node communication: N, a2a ( p) edgecut p N p Inverse local RAM bandwidth Local latency on working set n/p All to all remote bandwidth with p participating processors 12

149 BFS with a 2D partitioned graph Avoid expensive p way All to all communication step Each process collectively owns n/p r vertices Additional Allgather communication step for processes in a row Local memory references: m p n p L L, n p L, n c p r m p Inter node communication: N, a2a ( p N, gather r ( p edgecut ) p c ) 1 1 p r n p c N p r N p c 13

150 Temporal effects, communication minimizing tuning prevent us from obtaining tighter bounds The volume of communication can be further reduced by maintaining state of non local visited vertices 0 1 P0 [0,3] [0,3] [1,3] [0,4] [1,4] [0,6] [1,6] [1,6] Local pruning prior to All to all step 2 5 [0,3] [0,4] [1,6] 14

151 Predictable BFS execution time for synthetic small world graphs Randomly permuting vertex IDs ensures load balance on R MAT graphs (used in the Graph 500 benchmark). Our tuned parallel implementation for the NERSC Hopper system (Cray XE6) is ranked #2 on the current Graph 500 list. Execution time is dominated by work performed in a few parallel phases Buluc & Madduri, Parallel BFS on distributed memory systems, Proc. SC 11,

152 Modeling BFS execution time for real world graphs Can we further reduce communication time utilizing existing partitioning methods? Does the model predict execution time for arbitrary low diameter graphs? We try out various partitioning and graph distribution schemes on the DIMACS Challenge graph instances Natural ordering, Random, Metis, PaToH 16

153 Experimental Study The (weak) upper bound on aggregate data volume communication can be statically computed (based on partitioning of the graph) We determine runtime estimates of Total aggregate communication volume Sum of max. communication volume during each BFS iteration Intra node computational work balance Communication volume reduction with 2D partitioning We obtain and analyze execution times (at several different parallel concurrencies) on a Cray XE6 system (Hopper, NERSC) 17

154 Orderings for the CoPapersCiteseer graph Natural Random Metis PaToH PaToH checkerboard 18

155 BFS All to all phase total communication volume normalized to # of edges (m) Graph name Natural Random PaToH % compared to m # of partitions 19

156 Ratio of max. communication volume across iterations to total communication volume Graph name Natural Random PaToH Ratio over total volume # of partitions 20

157 Reduction in total All to all communication volume with 2D partitioning Graph name Natural Random PaToH Ratio compared to 1D # of partitions 21

158 Edge count balance with 2D partitioning Graph name Natural Random PaToH Max/Avg. ratio # of partitions

159 Parallel speedup on Hopper with 16 way partitioning 23

160 Comm. time (ms) Comm. time (ms) BFS time (ms) BFS time (ms) Execution time breakdown eu 2005 Computation Fold Expand Random 1D Random 2D Metis 1D PaToH 1D Partitioning Strategy Fold Expand Fold Expand Random 1D Random 2D Metis 1D PaToH 1D Partitioning Strategy kron simple logn18 Computation Fold Expand Random 1D Random 2D Metis 1D PaToH 1D Partitioning Strategy Random 1D Random 2D Metis 1D PaToH 1D Partitioning Strategy 24

161 Imbalance in parallel execution eu 2005, 16 processes* PaToH Random * Timeline of 4 processes shown in figures. PaToH partitioned graph suffers from severe load imbalance in computational phases. 25

162 Conclusions Randomly permuting vertex identifiers improves computational and communication load balance, particularly at higher process concurrencies Partitioning methods reduce overall communication volume, but introduce significant load imbalance Substantially lower parallel speedup with real world graphs compared to synthetic graphs (8.8X vs 50X at 256 way parallel concurrency) Points to the need for dynamic load balancing 26